Tagged Questions

Questions about the branch of combinatorics called graph theory (not to be used for questions concerning the graph of a function). This tag can be further specialized via using it in combination with more specialized tags such as extremal-graph-theory, spectral-graph-theory, algebraic-graph-theory, ...

For any graph $G=(V,E)$ let $\bar{G}$ be the complement graph. Is $$\text{inf}\big\{\frac{\omega(G)+\omega(\bar{G})}{\chi(G)} : G \text{ is a finite graph}\big\}$$ known? If not, what lower bounds are ...

While there exists regular tesselations of the hyperbolic plane with arbitrary regular polygons, there is no new regular polyhedra in hyperbolic (3D) space. This being quite trivial, is probably very ...

Let $k\geq2$ be an integer, a graph $G=(V,E)$ is called $k$-partite if $V$ admits a partition into $k$ classes such that every edge of $G$ has its ends in different classes: vertices in the same class ...

Given two graphs $G=(V_1,E_1)$ and $H=(V_2,E_2)$, the tensor product of $G$ and $H$ is the graph $G \times H = (V,E)$, where $V=V_1 \times V_2$ is the Cartesian product of the $V_i$ and
$ (u,v) \ E \ ...

Let $\lambda_m$ denote the $m$-th eigenvalue of the adjacency matrix $A$ of a bipartite graph $G$ of order $N$ and ${\bf v}_m$ the normalized eigenvector corresponding to $\lambda_m$. I am looking for ...

How we can prove this:
Suppose G is a connected graph with adjacent matrix A. prove that there exists k such that all elements of A^k are positive iff G is bipartite. Is A^n >0 if G is not bipartite?
...

The question started from a problem brought home by a friend's 5th grader: "How many ways you can arrange 5 people sitting around a round table so that the people sitting to the left of any person are ...

I was wondering whether there exists any kind of literature on the the powers of the discrete Laplacian, in particular the the discrete bi-Laplacian, possibly with weights on the edges.
In particular ...

Consider Thompson's group (the one commonly referred to as $T$), which is finitely presentable. Consider the Cayley graph, but then forget the coloring and direction on edges. So now we just have an ...

Let $G=(V,E)$ be a geometric graph, a graph embedded in the plane whose edge lengths are
the Euclidean distance between its endpoint vertices.
Say that a set of vertices $D \subseteq V$ is a geometric ...

I'm interested in the statistics of strongly connected components in random directed graphs. However, I'm unable to find any results on this, partly because I don't know the terminology to search for.
...

Could you point out some survey papers and monographs that highlight the kernel of tricks, techniques, and tools that Paul Erdős employed the most in his research work (in particular in graph theory, ...

For the purposes of this post, a digraph (directed graph) has neither loops nor multiple parallel edges, and a preset is an ordered pair consisting of a set $S$ and a preorder (viz., a reflexive and ...

All graphs discussed are finite and simple. The cycle sequence of a graph $G$, denoted $C(G)$, is the nondecreasing sequence of the lengths of all of the cycles in $G$, where cycles are distinguished ...

The only finite connected graphs $G$ that are isomorphic to their line graph $L(G)$ are the cycle graphs $C_n$ (see this link for example).
There are connected countable graphs that are isomorphic to ...

Consider a strongly regular graph $G$ with parameters $(76,30,8,14).$ Hoffman's bound tells us that $\overline{G}$ has an independent set of size at most $4$ and its not hard to see there are indeed ...